Oscillation of Nonlinear Fractional Dynamic Equations with Forcing Term

In this paper, interval oscillation criteria for the nonlinear damped dynamic equations with forcing terms on time scales within conformable fractional derivatives are established. Our approach is determined from the implementation of generalized Riccati transformation, some properties of conformable time-scale fractional calculus, and certain mathematical inequalities. Also, we extend the study of oscillation to conformable fractional Euler-type dynamic equation. Examples are presented to emphasize the validity of the main theorems\enleadertwodots.


Introduction
e consideration of dynamic equations on time scales has attracted many researchers because of its wide applications in the field of science and engineering. e theory of time scales was presented by Hilger [1] to unify the discrete and continuous analysis. It not only unifies the continuous and discrete cases but also gives new areas in between such as q-calculus [2]. e qualitative analysis of solutions of dynamic equations on different time scales has received considerable notice. In particular, the investigation of the oscillation of solutions to dynamic equations [3][4][5], dynamic equations with damping term [6][7][8], and dynamic equations on various time scales [9][10][11] has gained extensive attention. Fractional calculus is a generalization of integration and differentiation to any order. Recently, it has been realized that the fractional calculus has numerous applications in engineering, signal processing, economics and finance, probability and statistics, neural networks, and thermodynamics; see for illustrations [12][13][14][15][16] and the citations therein. In recent times, the importance has been given to fractional order calculus rather than integer order due to its applications in engineering such as neural networks, electrical and mechanical engineering, and population dynamics. e fractional dynamic equations on time scales have been studied by only few authors [17][18][19].
In [19], Feng and Meng established the asymptotic and oscillatory behavior of the following dynamic equation of fractional order on time scales using the generalized Riccati transformation technique: Alzabut et al. [17] considered the following nonlinear damped dynamic equation with a conformable fractional derivative: Here, the authors established the oscillation of the above equation when the nonlinear function f is increasing and nonincreasing. Besides, the results are carried out in light of the following two cases: Motivated by the above discussion, in this work, we established the oscillation results of nonlinear conformable fractional dynamic equations with a forced term.
In the recent papers [18,20], the conformable time-scale fractional calculus has been introduced. Applications of the obtained results demonstrate that the newly defined calculus will be applied to investigate oscillation for both fractional differential and fractional difference equations at the same time. erefore, the determination of oscillation of solutions of conformable fractional dynamic equations has become a promising topic for researchers. To the best of our observation, papers [17,19] are the only research that has studied the oscillation of conformable fractional dynamic equations.

Problem Formulation and Preliminaries
In this article, we are concerned with a class of nonlinear conformable fractional dynamic equations with damped and forced terms on time scales of the kind: where T denotes an arbitrary time scale, T , [0, ∞)), and f ∈ C(R, R) such that yf(y) > 0 and the function h(η) . We constitute new interval criteria for oscillation of the solutions of equation (3) when the nonlinear function f is increasing and nonincreasing and extend the results to the Euler-type fractional dynamic equations.
By a solution of (3), we insist a nontrivial function y(η) ∈ R fulfilling (3) for η ≥ η 0 . If a solution of (3) is neither eventually positive (EP) nor eventually negative (EN), then it is called oscillatory. Or else, it is said to be nonoscillatory. If all solutions of (3) are oscillatory, then (3) is called oscillatory.
Before we proceed to the main results, we present essential preliminaries on conformable time-scale fractional calculus that will be used to justify further discussion. Terms and definitions are adopted from the papers [2,18].
e graininess function μ(η) of the time scale is given by Definition 2 (see [2]). A real-valued function f defined on T is known as rd-continuous if at all left-dense points, a finite left limit of f exists, and if it is continuous at each right-dense point.
Theorem 1 (see [18]). By the definition of α-order con- Theorem 3 (see [18]). Let g, f be real-valued α differentiable functions defined on T at a point υ in T k . en, for a, b ∈ T.
For simplicity, we use the notion as follows:

Main Results
is part supplies the main theorems of the work. We will present the results in two folds based on the monotonicity of f and extend the results for fractional Euler-type dynamic equation. (3) has an EP solution y, then there is a sufficiently

Oscillation Criteria for (3) When f Is Not Necessarily
Increasing. To establish oscillation criteria, we make use of the following assumption: (H 1 )f(y)/y ≥ K ∀ y ≠ 0 and for some K > 0.
T and G ∈ G for sufficiently large T such that then (3) is oscillatory.
Proof. Assume that (3) has a nonoscillatory solution y. en, y is either EN or EP for η ≥ η 1 ≥ η 0 . Define a generalized Riccati function as follows: By Lemma 1, clearly w(η) ≥ 0 and 4 Journal of Mathematics By assumptions, we take α i , and [α 2 , β 2 ], the above inequality implies that (20) By multiplying G(η, υ) and taking α-fractional integral for (20) from c i to η, we have Journal of Mathematics 5 By multiplying G(υ, η) and taking integration of α-order from η to c i for (20), we have Dividing (33) by G(β i , c i ) and (24) by G(c i , α i ), we attain 6 Journal of Mathematics Adding the above two inequalities, we get a contradiction to hypothesis. □ Remark 1. Assume that the condition of eorem 5 holds for every T ≥ 0 and function G(υ, η) ∈ G for sufficiently large T. If there exist some c i ∈ (α i , β i ), i � 1, 2 so that for i � 1, 2, then (3) is oscillatory.
, and we use h(η − υ) for them. By using this G(η − υ) in eorem 5, we have the corollary as follows.

Corollary 1. Suppose that all the assumptions of eorem 5 hold and for every
. If there is a G: � G(η − υ) ∈ G such that then (3) is oscillatory.

Oscillation Criteria for (3) When f Is Increasing.
To establish oscillation criteria, we make use of the following assumption: (H2) f ′ exists and f ′ (y) ≥ M for all y ≠ 0 and for some M > 0 Theorem 6. Let Φ(η) ∈ R + . If (H 2 ), T and G ∈ G for sufficiently large T such that then (3) is oscillatory.